1.1 Overview and motivation

We have chosen the topic of crime because it has always been topical, as we can still see today because of the increasing number of crimes against humanity, hate crimes and violent acts around the world. Multiple wars in Europe and the Middle East; dubious governments in Africa; social networks that have become a digital court or a channel for harassment; or religious institutions like the Church that has been qualified as one of the biggest networks of sexual assault and pedophilia are just a few impressions that the situation is deteriorating year after year. But has this always been the case?

The U.S. is the world’s largest economic and military power but is still aggressively criticized for its various laws governing the right to bear arms, racism, poverty, and failing health and justice systems. The United States is no exception to the rule of increasing crime rates. Some cities, such as Houston in the state of Texas, is considered one of the most dangerous cities on American soil, where crime is out of control, where murders and trafficking of all kinds are almost freely carried out; a worrying case that reveals the particular situation of the United States regarding public safety, as stated in the article “Is Crime up or down ? In Houston, concerns are hard to allay” from the news agency “The Associated Press”.

As all three of us have a real interest in this country, the choice of making it the focus of our study seemed obvious. The United States is a federal state, which means that different autonomous entities run the country at their own level, so the states have their own constitution and laws. Because of its great political, cultural, and social diversity, the United States is the perfect choice for our study, which will be all the more accomplished thanks to the wealth of data.

Also, as explained in the article “Stories about crime are rife with misinformation and racism, critics say”, opinions are strongly divided regarding the information relayed by the media about crime. Some say that public opinion is highly manipulated by the government controlling the media, others claim the objectivity of the American media. For this reason, we would like to conduct our own study and better understand crime in the United States.

1.2 Related work

Here are some articles or studies that will guide our research:
- “What the data says (and doesn’t say) about crime in the United States”
- “What causes violent crime?”
- “Myths and Realities: Understanding Recent Trends in Violent Crime”

1.3 Research question

The purpose of this analysis is to estimate the impact of several macroeconomic variables on the crime rate in the United States. For this project we consider only violent crime, namely: homicide, rape, aggravated assault, and robbery. In addition, we consider the following macroeconomic variables: population, poverty rate, foreign-born persons residing in the United States, Police and Sheriff’s Patrol Officers, number of federal firearms licensees, income inequality, and unemployment rate. These variables are reported by state for the year 2021 (2020 for data that have not been updated for the year 2021).

We consider the following five questions as guidelines for our study:

1. Which states have high crime rates? We want to rank the states according to their crime rates, to identify potential similarities and/or differences, and to create groups that are representative of the entire country.

2. How have crime rates changed over time? Find out if states with high crime rates have always been high crime states, or if they have become high crime states (if so, understand why).

3. What are the common variables influencing crime rates? We will identify the variables that most impact the crime rate in each group by listing the most common variables potentially responsible for violent crime.

4. What impact do these different variables have on the crime rate? We would like to estimate the weight of each variable identified in question 2 on the crime rate.

5. How can we optimize the impact of these variables on crime? We would like to identify the potentially versatile variables, in order to propose feasible solutions to decrease the crime rate.

1.4 Data

2.1 Type of crimes dataset

2.2 Formation of State clusters

km.res[["centers"]]

##   Robbery_scaled Homicides_scaled Aggravated_assault_scaled Rape_scaled
## 1      0.7881724        1.4176379                1.48108088   0.4370596
## 2      1.1558337        1.0691436                2.75152733   4.5779411
## 3      0.8194608        0.4437583               -0.03728383  -0.5420362
## 4     -0.6686191       -0.8708740               -0.88735816  -0.3830740
## 5     -0.7665022       -0.3274591                0.21047178   0.7121579

2.3 Macroeconomic variables Dataset

2.4 Assumptions

3.1 Determine the robbery rate through the macro variables

We started by calculating the correlation matrix of this model:

corr_matrixRobb

We decided to calculate the following regression, which considers all the macro-variables presented above, to determine the robbery rate :

\(Robbery =\beta_0 + \beta_1* ForeignBorn21 + \beta_2 * WeaponsLicences + \beta_3 * UnemployementRate + \\\beta_4 * Officers + \beta_5 * PovRate +\beta_6 * IncomeInequal\)

We find an \(R^2\) of 0.5244, which means that 52% of the variation in the robbery rate is explained by the linear relationship with the macro variables.

By applying backward selection based on AIC, we are told that the Employees variable is rejected. Our final regression is then :

\(Robbery =\beta_0 + \beta_1* ForeignBorn21 + \beta_2 * WeaponsLicences + \beta_3 * UnemployementRate + \\ \beta_4 * PovRate +\beta_5 * IncomeInequal\)

summary(regRobbery)

## 
## Call:
## lm(formula = Robbery ~ Foreign_Born21 + Weapons_licences + Unemployement_rate + 
##     Pov_rate + Income_Inequal, data = Bigdata)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -35.093 -11.215   2.608  10.705  53.207 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)  
## (Intercept)        27.0311129 25.9726415   1.041   0.3038  
## Foreign_Born21      0.0014237  0.0008696   1.637   0.1089  
## Weapons_licences   -0.3090258  0.1292561  -2.391   0.0213 *
## Unemployement_rate  0.0081597  0.0037536   2.174   0.0353 *
## Pov_rate            0.0036082  0.0016160   2.233   0.0308 *
## Income_Inequal     -0.0032709  0.0022878  -1.430   0.1600  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 19.39 on 43 degrees of freedom
## Multiple R-squared:  0.5423, Adjusted R-squared:  0.4891 
## F-statistic: 10.19 on 5 and 43 DF,  p-value: 1.738e-06

We see that of the five variables considered, three of them are statistically significant. We therefore assume that Weapons licences, Unemployment rate and Poverty rate have a relevant impact on the determination of the robbery rate in the United States. Nevertheless, removing the variables that are not statistically different from 0 could change or cancel the impact of the others on the dependent variable. It is therefore important to keep all the variables that the backward induction suggested based on the AIC, as proven before.

We check if there is a multi-collinearity issue :

We have no VIF greater than 5, so we assume that multi-collinearity is not a concern.

We see that this model has an MAE of 14, which means that on average, the model has an error of 14. In terms of robberies, our model does not explain 1400000 of the robberies in the US.
We see that the points on the QQ-Plot do not fall on the curve but are not scattered far from it. There are therefore anomalies but the model remains relevant since there is a relationship between the independent variables and the dependent variable.

Our first hypothesis was that the robbery rate depended on the unemployment rate and income inequality, our model confirmed this hypothesis, however, the immigration rate, the number of firearms licenses issued and the poverty rate also influence the robbery rate.

3.2 Determine the Homicides rate through the macro variables

We started by calculating the correlation matrix of this model:

corr_matrixHomi

We decided to calculate the following regression, which considers all the macro-variables presented above, to determine the Homicides rate :

\(Homicides =\beta_0 + \beta_1* ForeignBorn21 + \beta_2 * WeaponsLicences + \beta_3 * UnemployementRate + \\ \beta_4 * Officers + \beta_5 * PovRate +\beta_6 * IncomeInequal\)

We find an \(R^2\) of 0.5811, which means that 58% of the variation in the theft rate is explained by the linear relationship with the macro variables.

By applying backward selection based on AIC, we are told that the Unemployement & Income Inequality variables are rejected. Our final regression is then :

\(Homicides =\beta_0 + \beta_1* ForeignBorn21 + \beta_2 * WeaponsLicences + \beta_3 * Officers + \\ \beta_4 * PovRate\)

summary(regHomicides)

## 
## Call:
## lm(formula = Homicides ~ Foreign_Born21 + Weapons_licences + 
##     Officers + Pov_rate, data = Bigdata)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.7030 -1.3072 -0.1916  1.3940  5.2785 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      -2.7163060  2.4968823  -1.088   0.2826    
## Foreign_Born21   -0.0001262  0.0000647  -1.951   0.0575 .  
## Weapons_licences -0.0320626  0.0136464  -2.350   0.0233 *  
## Officers          0.0174085  0.0098143   1.774   0.0830 .  
## Pov_rate          0.0006448  0.0001299   4.962 1.09e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.048 on 44 degrees of freedom
## Multiple R-squared:  0.5923, Adjusted R-squared:  0.5552 
## F-statistic: 15.98 on 4 and 44 DF,  p-value: 3.763e-08

We see that of the four variables included in the regression, weapons licenses and the poverty rate are statistically significant at a different level, meaning that varying them would have a large impact on the homicides rate. We notice that the intercept is negative for this regression and is equal to -2.7. It may seem worrying to have a negative intercept, but this is not a problem. It can be explained by the fact that our macro variables are expressed differently (some in tens, hundreds, or thousands) but are all reduced to 100k, the intercept will “correct” these differences.
We check if there is a multi-collinearity issue

We have no VIF greater than 5, so we assume that multi-collinearity is not a concern.

We see that the points on the QQ-Plot do not fall on the curve but are not scattered far from it. There are therefore anomalies but the model remains relevant since there is a relationship between the independent variables and the dependent variable.

Our second hypothesis said that the homicide rate depended on the immigration rate, the poverty rate, and the number of firearms licenses issued. Our regression model confirmed this hypothesis, but the rate of law enforcement officers also impacts the homicide rate.

3.3 Determine the Aggravated Assault rate through the macro variables

We started by calculating the correlation matrix of this model:

corr_matrixAssault

We decided to calculate the following regression, which considers all the macro-variables presented above, to determine the Aggravated_assault rate :

\(Assault =\beta_0 + \beta_1* ForeignBorn21 + \beta_2 * WeaponsLicences + \beta_3 * UnemployementRate + \\\beta_4 * Officers + \beta_5 * PovRate +\beta_6 * IncomeInequal\)

We find an \(R^2\) of 0.3373, which means that 34% of the variation in the theft rate is explained by the linear relationship with the macro variables.

By applying backward selection based on AIC, we are told that all the variables are rejected except Poverty rate. Our final regression is then :

\(Assault =\beta_0 + \ \beta_1 * PovRate\)

summary(regAssault)

## 
## Call:
## lm(formula = Aggravated_assault ~ Pov_rate, data = Bigdata)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -262.716  -64.328   -4.949   34.947  244.526 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -71.908152  68.558458  -1.049      0.3    
## Pov_rate      0.027249   0.005334   5.109 5.81e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 98.31 on 47 degrees of freedom
## Multiple R-squared:  0.3571, Adjusted R-squared:  0.3434 
## F-statistic:  26.1 on 1 and 47 DF,  p-value: 5.815e-06

We note that five clearly identifiable points stand out on the red curve. Again, some states have lower than average rates of aggravated assault (Maine, New Hampshire, Connecticut), others higher (New Mexico, Tennessee). The rest of the points are very close to the curve.

Our third hypothesis was that the rate of aggravated assaults depended on the number of firearms licenses issued and the unemployment rate. This was disproved by our model, which assumes that the homicide rate is only affected by the poverty rate.

3.4 Determine the Rape rate through the macro variables

We started by calculating the correlation matrix of this model:

corr_matrixRape

We decided to calculate the following regression, which considers all the macro-variables presented above, to determine the Rape rate :

\(Rape =\beta_0 + \beta_1* ForeignBorn21 + \beta_2 * WeaponsLicences + \beta_3 * UnemployementRate + \\ \beta_4 * Officers + \beta_5 * PovRate +\beta_6 * IncomeInequal\)

We find an \(R^2\) of 0.4257, which means that 42% of the variation in the theft rate is explained by the linear relationship with the macro variables.

By applying backward selection based on AIC, we are told that ForeignBorn & Officers variables are rejected. Our final regression is then :

\(Rape =\beta_0 + \beta_1 * WeaponsLicences + \beta_2 * UnemployementRate + \\ \beta_3 * PovRate +\beta_4 * IncomeInequal\)

summary(regRape)

## 
## Call:
## lm(formula = Rape_net ~ Weapons_licences + Unemployement_rate + 
##     Pov_rate + Income_Inequal, data = Bigdata)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16.2484  -7.2032  -0.8501   3.7470  24.2894 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        49.3576978 13.5252639   3.649 0.000694 ***
## Weapons_licences    0.1571027  0.0598584   2.625 0.011884 *  
## Unemployement_rate  0.0017767  0.0017675   1.005 0.320311    
## Pov_rate            0.0021888  0.0006554   3.340 0.001717 ** 
## Income_Inequal     -0.0032506  0.0011215  -2.898 0.005828 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.16 on 44 degrees of freedom
## Multiple R-squared:  0.4023, Adjusted R-squared:  0.3479 
## F-statistic: 7.403 on 4 and 44 DF,  p-value: 0.0001193

Regarding the final regression of the rape rate, three of the four variables considered are statistically significant at different levels: weapons licensees, poverty rate and income inequality, so they have an influence on the rape rate.

We check if there is a multi-collinearity issue :

We have no VIF greater than 5, so we assume that multi-collinearity is not a concern.

We can see that the QQ-plot has some anomalies, related to states with higher than average rape rates, like Arkansas, South Dakota, Colorado, Michigan… The other points are on the curve or very close.

Our fourth hypothesis said that the rape rate depended on the number of firearms licenses issued and the rate of income inequality. Our model confirmed this hypothesis, but the unemployment rate and the poverty rate also affect the rape rate.